On equivalent characterizations of weakly compactly generated Banach spaces (Q1890813)

Let us consider the following nice Theorem. For a Banach space \(V\) the following assertions are equivalent: (a) \(V\) is weakly compactly generated (w.c.g.), (b) \(V\) is GSG and simultaneously a Vašák (i.e., weakly \(K\)-countable determined) space, and (c) \(V\) is GSG and moreover \((V^*,w^*)\) continuously injects into \(\Sigma(\Gamma)\) for some set \(\Gamma\). A main aim of this paper is to present a more direct proof that (b) or (c) implies (a). We shall avoid interpolation as well as gymnastics involving Gul'ko and Corson compacta. In particular, we shall no longer need a result of Gul'ko that a continuous image of a Corson compact is a Corson compact. A central concept, we shall use in our proof will be a slight variant of a projectional generator introduced recently by \textit{J. Orihuela} and \textit{M. Valdivia} [Rev. Math. Univ. Complutense Madrid 2, 179-199 (1989; Zbl 0717.46009)].